LMFDB - Embedded newform 3120.2.a.bj.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3120,2,Mod(1,3120)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3120, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("3120.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3120.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$24.9133254306$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	3120.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.00000 q^{5} +1.19656 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.00000 q^{5} +1.19656 q^{7} +1.00000 q^{9} +1.19656 q^{11} +1.00000 q^{13} -1.00000 q^{15} +6.17513 q^{17} -6.97858 q^{19} +1.19656 q^{21} -4.17513 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +2.97858 q^{31} +1.19656 q^{33} -1.19656 q^{35} +7.78202 q^{37} +1.00000 q^{39} -6.17513 q^{41} +9.95715 q^{43} -1.00000 q^{45} +1.02142 q^{47} -5.56825 q^{49} +6.17513 q^{51} +10.1751 q^{53} -1.19656 q^{55} -6.97858 q^{57} -5.37169 q^{59} +12.5682 q^{61} +1.19656 q^{63} -1.00000 q^{65} -9.37169 q^{67} -4.17513 q^{69} +5.19656 q^{71} -11.9572 q^{73} +1.00000 q^{75} +1.43175 q^{77} +1.78202 q^{79} +1.00000 q^{81} +5.37169 q^{83} -6.17513 q^{85} +6.00000 q^{87} +10.1751 q^{89} +1.19656 q^{91} +2.97858 q^{93} +6.97858 q^{95} -1.82487 q^{97} +1.19656 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^5 - q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - 3 q^{15} - q^{17} - 6 q^{19} - q^{21} + 7 q^{23} + 3 q^{25} + 3 q^{27} + 18 q^{29} - 6 q^{31} - q^{33} + q^{35} + 13 q^{37} + 3 q^{39} + q^{41} - 3 q^{45} + 18 q^{47} + 12 q^{49} - q^{51} + 11 q^{53} + q^{55} - 6 q^{57} + 8 q^{59} + 9 q^{61} - q^{63} - 3 q^{65} - 4 q^{67} + 7 q^{69} + 11 q^{71} - 6 q^{73} + 3 q^{75} + 33 q^{77} - 5 q^{79} + 3 q^{81} - 8 q^{83} + q^{85} + 18 q^{87} + 11 q^{89} - q^{91} - 6 q^{93} + 6 q^{95} - 25 q^{97} - q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^5 - q^7 + 3 * q^9 - q^11 + 3 * q^13 - 3 * q^15 - q^17 - 6 * q^19 - q^21 + 7 * q^23 + 3 * q^25 + 3 * q^27 + 18 * q^29 - 6 * q^31 - q^33 + q^35 + 13 * q^37 + 3 * q^39 + q^41 - 3 * q^45 + 18 * q^47 + 12 * q^49 - q^51 + 11 * q^53 + q^55 - 6 * q^57 + 8 * q^59 + 9 * q^61 - q^63 - 3 * q^65 - 4 * q^67 + 7 * q^69 + 11 * q^71 - 6 * q^73 + 3 * q^75 + 33 * q^77 - 5 * q^79 + 3 * q^81 - 8 * q^83 + q^85 + 18 * q^87 + 11 * q^89 - q^91 - 6 * q^93 + 6 * q^95 - 25 * q^97 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.19656	0.452256	0.226128	−	0.974098i	$-0.427393\pi$
$7$	1.19656	0.452256	0.226128	+	0.974098i	$0.427393\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.19656	0.360776	0.180388	−	0.983596i	$-0.442265\pi$
$11$	1.19656	0.360776	0.180388	+	0.983596i	$0.442265\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−1.00000	−0.258199
$16$	0	0
$17$	6.17513	1.49769	0.748845	−	0.662745i	$-0.230609\pi$
$17$	6.17513	1.49769	0.748845	+	0.662745i	$0.230609\pi$
$18$	0	0
$19$	−6.97858	−1.60100	−0.800498	−	0.599336i	$-0.795431\pi$
$19$	−6.97858	−1.60100	−0.800498	+	0.599336i	$0.795431\pi$
$20$	0	0
$21$	1.19656	0.261110
$22$	0	0
$23$	−4.17513	−0.870576	−0.435288	−	0.900291i	$-0.643353\pi$
$23$	−4.17513	−0.870576	−0.435288	+	0.900291i	$0.643353\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	2.97858	0.534968	0.267484	−	0.963562i	$-0.413808\pi$
$31$	2.97858	0.534968	0.267484	+	0.963562i	$0.413808\pi$
$32$	0	0
$33$	1.19656	0.208294
$34$	0	0
$35$	−1.19656	−0.202255
$36$	0	0
$37$	7.78202	1.27936	0.639678	−	0.768643i	$-0.279068\pi$
$37$	7.78202	1.27936	0.639678	+	0.768643i	$0.279068\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−6.17513	−0.964394	−0.482197	−	0.876063i	$-0.660161\pi$
$41$	−6.17513	−0.964394	−0.482197	+	0.876063i	$0.660161\pi$
$42$	0	0
$43$	9.95715	1.51845	0.759226	−	0.650827i	$-0.225578\pi$
$43$	9.95715	1.51845	0.759226	+	0.650827i	$0.225578\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	1.02142	0.148990	0.0744949	−	0.997221i	$-0.476266\pi$
$47$	1.02142	0.148990	0.0744949	+	0.997221i	$0.476266\pi$
$48$	0	0
$49$	−5.56825	−0.795464
$50$	0	0
$51$	6.17513	0.864692
$52$	0	0
$53$	10.1751	1.39766	0.698831	−	0.715287i	$-0.253704\pi$
$53$	10.1751	1.39766	0.698831	+	0.715287i	$0.253704\pi$
$54$	0	0
$55$	−1.19656	−0.161344
$56$	0	0
$57$	−6.97858	−0.924335
$58$	0	0
$59$	−5.37169	−0.699335	−0.349667	−	0.936874i	$-0.613705\pi$
$59$	−5.37169	−0.699335	−0.349667	+	0.936874i	$0.613705\pi$
$60$	0	0
$61$	12.5682	1.60920	0.804600	−	0.593818i	$-0.202380\pi$
$61$	12.5682	1.60920	0.804600	+	0.593818i	$0.202380\pi$
$62$	0	0
$63$	1.19656	0.150752
$64$	0	0
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−9.37169	−1.14493	−0.572467	−	0.819928i	$-0.694014\pi$
$67$	−9.37169	−1.14493	−0.572467	+	0.819928i	$0.694014\pi$
$68$	0	0
$69$	−4.17513	−0.502627
$70$	0	0
$71$	5.19656	0.616718	0.308359	−	0.951270i	$-0.400220\pi$
$71$	5.19656	0.616718	0.308359	+	0.951270i	$0.400220\pi$
$72$	0	0
$73$	−11.9572	−1.39948	−0.699740	−	0.714398i	$-0.746701\pi$
$73$	−11.9572	−1.39948	−0.699740	+	0.714398i	$0.746701\pi$
$74$	0	0
$75$	1.00000	0.115470
$76$	0	0
$77$	1.43175	0.163163
$78$	0	0
$79$	1.78202	0.200493	0.100246	−	0.994963i	$-0.468037\pi$
$79$	1.78202	0.200493	0.100246	+	0.994963i	$0.468037\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.37169	0.589620	0.294810	−	0.955556i	$-0.404744\pi$
$83$	5.37169	0.589620	0.294810	+	0.955556i	$0.404744\pi$
$84$	0	0
$85$	−6.17513	−0.669787
$86$	0	0
$87$	6.00000	0.643268
$88$	0	0
$89$	10.1751	1.07856	0.539281	−	0.842126i	$-0.318696\pi$
$89$	10.1751	1.07856	0.539281	+	0.842126i	$0.318696\pi$
$90$	0	0
$91$	1.19656	0.125433
$92$	0	0
$93$	2.97858	0.308864
$94$	0	0
$95$	6.97858	0.715987
$96$	0	0
$97$	−1.82487	−0.185287	−0.0926435	−	0.995699i	$-0.529532\pi$
$97$	−1.82487	−0.185287	−0.0926435	+	0.995699i	$0.529532\pi$
$98$	0	0
$99$	1.19656	0.120259
$100$	0	0
$101$	−10.3503	−1.02989	−0.514945	−	0.857223i	$-0.672188\pi$
$101$	−10.3503	−1.02989	−0.514945	+	0.857223i	$0.672188\pi$
$102$	0	0
$103$	−18.7434	−1.84684	−0.923420	−	0.383790i	$-0.874619\pi$
$103$	−18.7434	−1.84684	−0.923420	+	0.383790i	$0.874619\pi$
$104$	0	0
$105$	−1.19656	−0.116772
$106$	0	0
$107$	18.5682	1.79506	0.897530	−	0.440953i	$-0.145359\pi$
$107$	18.5682	1.79506	0.897530	+	0.440953i	$0.145359\pi$
$108$	0	0
$109$	8.39312	0.803915	0.401957	−	0.915658i	$-0.368330\pi$
$109$	8.39312	0.803915	0.401957	+	0.915658i	$0.368330\pi$
$110$	0	0
$111$	7.78202	0.738637
$112$	0	0
$113$	7.95715	0.748546	0.374273	−	0.927319i	$-0.377892\pi$
$113$	7.95715	0.748546	0.374273	+	0.927319i	$0.377892\pi$
$114$	0	0
$115$	4.17513	0.389333
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	7.38890	0.677340
$120$	0	0
$121$	−9.56825	−0.869841
$122$	0	0
$123$	−6.17513	−0.556793
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	10.3931	0.922240	0.461120	−	0.887338i	$-0.347448\pi$
$127$	10.3931	0.922240	0.461120	+	0.887338i	$0.347448\pi$
$128$	0	0
$129$	9.95715	0.876679
$130$	0	0
$131$	6.39312	0.558569	0.279285	−	0.960208i	$-0.409903\pi$
$131$	6.39312	0.558569	0.279285	+	0.960208i	$0.409903\pi$
$132$	0	0
$133$	−8.35027	−0.724060
$134$	0	0
$135$	−1.00000	−0.0860663
$136$	0	0
$137$	16.7434	1.43048	0.715242	−	0.698877i	$-0.246316\pi$
$137$	16.7434	1.43048	0.715242	+	0.698877i	$0.246316\pi$
$138$	0	0
$139$	−5.78202	−0.490424	−0.245212	−	0.969469i	$-0.578858\pi$
$139$	−5.78202	−0.490424	−0.245212	+	0.969469i	$0.578858\pi$
$140$	0	0
$141$	1.02142	0.0860193
$142$	0	0
$143$	1.19656	0.100061
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	−5.56825	−0.459262
$148$	0	0
$149$	15.3461	1.25720	0.628599	−	0.777730i	$-0.283629\pi$
$149$	15.3461	1.25720	0.628599	+	0.777730i	$0.283629\pi$
$150$	0	0
$151$	8.58546	0.698675	0.349337	−	0.936997i	$-0.386407\pi$
$151$	8.58546	0.698675	0.349337	+	0.936997i	$0.386407\pi$
$152$	0	0
$153$	6.17513	0.499230
$154$	0	0
$155$	−2.97858	−0.239245
$156$	0	0
$157$	2.78623	0.222365	0.111183	−	0.993800i	$-0.464536\pi$
$157$	2.78623	0.222365	0.111183	+	0.993800i	$0.464536\pi$
$158$	0	0
$159$	10.1751	0.806941
$160$	0	0
$161$	−4.99579	−0.393723
$162$	0	0
$163$	−8.76060	−0.686183	−0.343091	−	0.939302i	$-0.611474\pi$
$163$	−8.76060	−0.686183	−0.343091	+	0.939302i	$0.611474\pi$
$164$	0	0
$165$	−1.19656	−0.0931519
$166$	0	0
$167$	17.3717	1.34426	0.672131	−	0.740432i	$-0.265379\pi$
$167$	17.3717	1.34426	0.672131	+	0.740432i	$0.265379\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−6.97858	−0.533665
$172$	0	0
$173$	−7.95715	−0.604971	−0.302486	−	0.953154i	$-0.597816\pi$
$173$	−7.95715	−0.604971	−0.302486	+	0.953154i	$0.597816\pi$
$174$	0	0
$175$	1.19656	0.0904513
$176$	0	0
$177$	−5.37169	−0.403761
$178$	0	0
$179$	15.5640	1.16331	0.581655	−	0.813435i	$-0.302405\pi$
$179$	15.5640	1.16331	0.581655	+	0.813435i	$0.302405\pi$
$180$	0	0
$181$	15.7820	1.17307	0.586534	−	0.809925i	$-0.300492\pi$
$181$	15.7820	1.17307	0.586534	+	0.809925i	$0.300492\pi$
$182$	0	0
$183$	12.5682	0.929072
$184$	0	0
$185$	−7.78202	−0.572145
$186$	0	0
$187$	7.38890	0.540330
$188$	0	0
$189$	1.19656	0.0870368
$190$	0	0
$191$	−10.7434	−0.777364	−0.388682	−	0.921372i	$-0.627070\pi$
$191$	−10.7434	−0.777364	−0.388682	+	0.921372i	$0.627070\pi$
$192$	0	0
$193$	9.73917	0.701041	0.350521	−	0.936555i	$-0.386005\pi$
$193$	9.73917	0.701041	0.350521	+	0.936555i	$0.386005\pi$
$194$	0	0
$195$	−1.00000	−0.0716115
$196$	0	0
$197$	−9.56404	−0.681410	−0.340705	−	0.940170i	$-0.610666\pi$
$197$	−9.56404	−0.681410	−0.340705	+	0.940170i	$0.610666\pi$
$198$	0	0
$199$	−5.95715	−0.422291	−0.211146	−	0.977455i	$-0.567719\pi$
$199$	−5.95715	−0.422291	−0.211146	+	0.977455i	$0.567719\pi$
$200$	0	0
$201$	−9.37169	−0.661028
$202$	0	0
$203$	7.17935	0.503891
$204$	0	0
$205$	6.17513	0.431290
$206$	0	0
$207$	−4.17513	−0.290192
$208$	0	0
$209$	−8.35027	−0.577600
$210$	0	0
$211$	−23.9143	−1.64633	−0.823164	−	0.567803i	$-0.807794\pi$
$211$	−23.9143	−1.64633	−0.823164	+	0.567803i	$0.807794\pi$
$212$	0	0
$213$	5.19656	0.356062
$214$	0	0
$215$	−9.95715	−0.679072
$216$	0	0
$217$	3.56404	0.241943
$218$	0	0
$219$	−11.9572	−0.807990
$220$	0	0
$221$	6.17513	0.415385
$222$	0	0
$223$	−2.62831	−0.176004	−0.0880022	−	0.996120i	$-0.528048\pi$
$223$	−2.62831	−0.176004	−0.0880022	+	0.996120i	$0.528048\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	0	0
$227$	−15.7648	−1.04635	−0.523174	−	0.852226i	$-0.675252\pi$
$227$	−15.7648	−1.04635	−0.523174	+	0.852226i	$0.675252\pi$
$228$	0	0
$229$	8.74338	0.577779	0.288890	−	0.957362i	$-0.406714\pi$
$229$	8.74338	0.577779	0.288890	+	0.957362i	$0.406714\pi$
$230$	0	0
$231$	1.43175	0.0942022
$232$	0	0
$233$	−2.17513	−0.142498	−0.0712489	−	0.997459i	$-0.522698\pi$
$233$	−2.17513	−0.142498	−0.0712489	+	0.997459i	$0.522698\pi$
$234$	0	0
$235$	−1.02142	−0.0666303
$236$	0	0
$237$	1.78202	0.115755
$238$	0	0
$239$	−2.80344	−0.181340	−0.0906698	−	0.995881i	$-0.528901\pi$
$239$	−2.80344	−0.181340	−0.0906698	+	0.995881i	$0.528901\pi$
$240$	0	0
$241$	−6.00000	−0.386494	−0.193247	−	0.981150i	$-0.561902\pi$
$241$	−6.00000	−0.386494	−0.193247	+	0.981150i	$0.561902\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	5.56825	0.355742
$246$	0	0
$247$	−6.97858	−0.444036
$248$	0	0
$249$	5.37169	0.340417
$250$	0	0
$251$	23.9143	1.50946	0.754729	−	0.656037i	$-0.227768\pi$
$251$	23.9143	1.50946	0.754729	+	0.656037i	$0.227768\pi$
$252$	0	0
$253$	−4.99579	−0.314083
$254$	0	0
$255$	−6.17513	−0.386702
$256$	0	0
$257$	−19.9572	−1.24489	−0.622447	−	0.782662i	$-0.713861\pi$
$257$	−19.9572	−1.24489	−0.622447	+	0.782662i	$0.713861\pi$
$258$	0	0
$259$	9.31163	0.578597
$260$	0	0
$261$	6.00000	0.371391
$262$	0	0
$263$	−8.00000	−0.493301	−0.246651	−	0.969104i	$-0.579330\pi$
$263$	−8.00000	−0.493301	−0.246651	+	0.969104i	$0.579330\pi$
$264$	0	0
$265$	−10.1751	−0.625054
$266$	0	0
$267$	10.1751	0.622708
$268$	0	0
$269$	−2.35027	−0.143298	−0.0716492	−	0.997430i	$-0.522826\pi$
$269$	−2.35027	−0.143298	−0.0716492	+	0.997430i	$0.522826\pi$
$270$	0	0
$271$	−10.9786	−0.666901	−0.333451	−	0.942768i	$-0.608213\pi$
$271$	−10.9786	−0.666901	−0.333451	+	0.942768i	$0.608213\pi$
$272$	0	0
$273$	1.19656	0.0724190
$274$	0	0
$275$	1.19656	0.0721551
$276$	0	0
$277$	1.21377	0.0729283	0.0364642	−	0.999335i	$-0.488391\pi$
$277$	1.21377	0.0729283	0.0364642	+	0.999335i	$0.488391\pi$
$278$	0	0
$279$	2.97858	0.178323
$280$	0	0
$281$	11.9572	0.713304	0.356652	−	0.934237i	$-0.383918\pi$
$281$	11.9572	0.713304	0.356652	+	0.934237i	$0.383918\pi$
$282$	0	0
$283$	29.8715	1.77567	0.887837	−	0.460158i	$-0.152207\pi$
$283$	29.8715	1.77567	0.887837	+	0.460158i	$0.152207\pi$
$284$	0	0
$285$	6.97858	0.413375
$286$	0	0
$287$	−7.38890	−0.436153
$288$	0	0
$289$	21.1323	1.24308
$290$	0	0
$291$	−1.82487	−0.106975
$292$	0	0
$293$	0.777809	0.0454401	0.0227200	−	0.999742i	$-0.492767\pi$
$293$	0.777809	0.0454401	0.0227200	+	0.999742i	$0.492767\pi$
$294$	0	0
$295$	5.37169	0.312752
$296$	0	0
$297$	1.19656	0.0694313
$298$	0	0
$299$	−4.17513	−0.241454
$300$	0	0
$301$	11.9143	0.686729
$302$	0	0
$303$	−10.3503	−0.594607
$304$	0	0
$305$	−12.5682	−0.719656
$306$	0	0
$307$	−0.760597	−0.0434095	−0.0217048	−	0.999764i	$-0.506909\pi$
$307$	−0.760597	−0.0434095	−0.0217048	+	0.999764i	$0.506909\pi$
$308$	0	0
$309$	−18.7434	−1.06627
$310$	0	0
$311$	23.1281	1.31147	0.655736	−	0.754990i	$-0.272358\pi$
$311$	23.1281	1.31147	0.655736	+	0.754990i	$0.272358\pi$
$312$	0	0
$313$	−33.9143	−1.91695	−0.958475	−	0.285176i	$-0.907948\pi$
$313$	−33.9143	−1.91695	−0.958475	+	0.285176i	$0.907948\pi$
$314$	0	0
$315$	−1.19656	−0.0674184
$316$	0	0
$317$	9.64973	0.541983	0.270991	−	0.962582i	$-0.412648\pi$
$317$	9.64973	0.541983	0.270991	+	0.962582i	$0.412648\pi$
$318$	0	0
$319$	7.17935	0.401966
$320$	0	0
$321$	18.5682	1.03638
$322$	0	0
$323$	−43.0937	−2.39780
$324$	0	0
$325$	1.00000	0.0554700
$326$	0	0
$327$	8.39312	0.464140
$328$	0	0
$329$	1.22219	0.0673816
$330$	0	0
$331$	−15.3288	−0.842550	−0.421275	−	0.906933i	$-0.638417\pi$
$331$	−15.3288	−0.842550	−0.421275	+	0.906933i	$0.638417\pi$
$332$	0	0
$333$	7.78202	0.426452
$334$	0	0
$335$	9.37169	0.512030
$336$	0	0
$337$	−22.3503	−1.21750	−0.608748	−	0.793363i	$-0.708328\pi$
$337$	−22.3503	−1.21750	−0.608748	+	0.793363i	$0.708328\pi$
$338$	0	0
$339$	7.95715	0.432173
$340$	0	0
$341$	3.56404	0.193004
$342$	0	0
$343$	−15.0386	−0.812010
$344$	0	0
$345$	4.17513	0.224782
$346$	0	0
$347$	5.78202	0.310395	0.155198	−	0.987883i	$-0.450399\pi$
$347$	5.78202	0.310395	0.155198	+	0.987883i	$0.450399\pi$
$348$	0	0
$349$	−27.5212	−1.47318	−0.736588	−	0.676342i	$-0.763564\pi$
$349$	−27.5212	−1.47318	−0.736588	+	0.676342i	$0.763564\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−28.7434	−1.52986	−0.764928	−	0.644116i	$-0.777225\pi$
$353$	−28.7434	−1.52986	−0.764928	+	0.644116i	$0.777225\pi$
$354$	0	0
$355$	−5.19656	−0.275805
$356$	0	0
$357$	7.38890	0.391062
$358$	0	0
$359$	12.5855	0.664235	0.332118	−	0.943238i	$-0.392237\pi$
$359$	12.5855	0.664235	0.332118	+	0.943238i	$0.392237\pi$
$360$	0	0
$361$	29.7005	1.56319
$362$	0	0
$363$	−9.56825	−0.502203
$364$	0	0
$365$	11.9572	0.625866
$366$	0	0
$367$	27.9143	1.45712	0.728558	−	0.684985i	$-0.240191\pi$
$367$	27.9143	1.45712	0.728558	+	0.684985i	$0.240191\pi$
$368$	0	0
$369$	−6.17513	−0.321465
$370$	0	0
$371$	12.1751	0.632102
$372$	0	0
$373$	−2.35027	−0.121692	−0.0608462	−	0.998147i	$-0.519380\pi$
$373$	−2.35027	−0.121692	−0.0608462	+	0.998147i	$0.519380\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	0	0
$377$	6.00000	0.309016
$378$	0	0
$379$	−24.5510	−1.26110	−0.630551	−	0.776148i	$-0.717171\pi$
$379$	−24.5510	−1.26110	−0.630551	+	0.776148i	$0.717171\pi$
$380$	0	0
$381$	10.3931	0.532455
$382$	0	0
$383$	−5.80765	−0.296757	−0.148379	−	0.988931i	$-0.547405\pi$
$383$	−5.80765	−0.296757	−0.148379	+	0.988931i	$0.547405\pi$
$384$	0	0
$385$	−1.43175	−0.0729687
$386$	0	0
$387$	9.95715	0.506151
$388$	0	0
$389$	13.6497	0.692069	0.346034	−	0.938222i	$-0.387528\pi$
$389$	13.6497	0.692069	0.346034	+	0.938222i	$0.387528\pi$
$390$	0	0
$391$	−25.7820	−1.30385
$392$	0	0
$393$	6.39312	0.322490
$394$	0	0
$395$	−1.78202	−0.0896631
$396$	0	0
$397$	−12.1323	−0.608902	−0.304451	−	0.952528i	$-0.598473\pi$
$397$	−12.1323	−0.608902	−0.304451	+	0.952528i	$0.598473\pi$
$398$	0	0
$399$	−8.35027	−0.418036
$400$	0	0
$401$	−37.4439	−1.86986	−0.934930	−	0.354832i	$-0.884538\pi$
$401$	−37.4439	−1.86986	−0.934930	+	0.354832i	$0.884538\pi$
$402$	0	0
$403$	2.97858	0.148373
$404$	0	0
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	9.31163	0.461561
$408$	0	0
$409$	−14.0000	−0.692255	−0.346128	−	0.938187i	$-0.612504\pi$
$409$	−14.0000	−0.692255	−0.346128	+	0.938187i	$0.612504\pi$
$410$	0	0
$411$	16.7434	0.825890
$412$	0	0
$413$	−6.42754	−0.316279
$414$	0	0
$415$	−5.37169	−0.263686
$416$	0	0
$417$	−5.78202	−0.283147
$418$	0	0
$419$	−3.17935	−0.155321	−0.0776606	−	0.996980i	$-0.524745\pi$
$419$	−3.17935	−0.155321	−0.0776606	+	0.996980i	$0.524745\pi$
$420$	0	0
$421$	−16.3074	−0.794775	−0.397388	−	0.917651i	$-0.630083\pi$
$421$	−16.3074	−0.794775	−0.397388	+	0.917651i	$0.630083\pi$
$422$	0	0
$423$	1.02142	0.0496633
$424$	0	0
$425$	6.17513	0.299538
$426$	0	0
$427$	15.0386	0.727771
$428$	0	0
$429$	1.19656	0.0577703
$430$	0	0
$431$	−4.58546	−0.220874	−0.110437	−	0.993883i	$-0.535225\pi$
$431$	−4.58546	−0.220874	−0.110437	+	0.993883i	$0.535225\pi$
$432$	0	0
$433$	−38.3503	−1.84300	−0.921498	−	0.388383i	$-0.873034\pi$
$433$	−38.3503	−1.84300	−0.921498	+	0.388383i	$0.873034\pi$
$434$	0	0
$435$	−6.00000	−0.287678
$436$	0	0
$437$	29.1365	1.39379
$438$	0	0
$439$	−7.73917	−0.369371	−0.184685	−	0.982798i	$-0.559127\pi$
$439$	−7.73917	−0.369371	−0.184685	+	0.982798i	$0.559127\pi$
$440$	0	0
$441$	−5.56825	−0.265155
$442$	0	0
$443$	−34.9185	−1.65903	−0.829514	−	0.558485i	$-0.811383\pi$
$443$	−34.9185	−1.65903	−0.829514	+	0.558485i	$0.811383\pi$
$444$	0	0
$445$	−10.1751	−0.482348
$446$	0	0
$447$	15.3461	0.725844
$448$	0	0
$449$	−6.17513	−0.291423	−0.145711	−	0.989327i	$-0.546547\pi$
$449$	−6.17513	−0.291423	−0.145711	+	0.989327i	$0.546547\pi$
$450$	0	0
$451$	−7.38890	−0.347930
$452$	0	0
$453$	8.58546	0.403380
$454$	0	0
$455$	−1.19656	−0.0560955
$456$	0	0
$457$	1.38890	0.0649702	0.0324851	−	0.999472i	$-0.489658\pi$
$457$	1.38890	0.0649702	0.0324851	+	0.999472i	$0.489658\pi$
$458$	0	0
$459$	6.17513	0.288231
$460$	0	0
$461$	28.4826	1.32657	0.663283	−	0.748369i	$-0.269163\pi$
$461$	28.4826	1.32657	0.663283	+	0.748369i	$0.269163\pi$
$462$	0	0
$463$	−16.3759	−0.761053	−0.380526	−	0.924770i	$-0.624257\pi$
$463$	−16.3759	−0.761053	−0.380526	+	0.924770i	$0.624257\pi$
$464$	0	0
$465$	−2.97858	−0.138128
$466$	0	0
$467$	−25.7476	−1.19146	−0.595728	−	0.803186i	$-0.703136\pi$
$467$	−25.7476	−1.19146	−0.595728	+	0.803186i	$0.703136\pi$
$468$	0	0
$469$	−11.2138	−0.517804
$470$	0	0
$471$	2.78623	0.128383
$472$	0	0
$473$	11.9143	0.547820
$474$	0	0
$475$	−6.97858	−0.320199
$476$	0	0
$477$	10.1751	0.465887
$478$	0	0
$479$	7.58967	0.346781	0.173391	−	0.984853i	$-0.444528\pi$
$479$	7.58967	0.346781	0.173391	+	0.984853i	$0.444528\pi$
$480$	0	0
$481$	7.78202	0.354830
$482$	0	0
$483$	−4.99579	−0.227316
$484$	0	0
$485$	1.82487	0.0828629
$486$	0	0
$487$	−4.41033	−0.199851	−0.0999255	−	0.994995i	$-0.531860\pi$
$487$	−4.41033	−0.199851	−0.0999255	+	0.994995i	$0.531860\pi$
$488$	0	0
$489$	−8.76060	−0.396168
$490$	0	0
$491$	0.0856914	0.00386720	0.00193360	−	0.999998i	$-0.499385\pi$
$491$	0.0856914	0.00386720	0.00193360	+	0.999998i	$0.499385\pi$
$492$	0	0
$493$	37.0508	1.66868
$494$	0	0
$495$	−1.19656	−0.0537813
$496$	0	0
$497$	6.21798	0.278915
$498$	0	0
$499$	17.7220	0.793344	0.396672	−	0.917960i	$-0.370165\pi$
$499$	17.7220	0.793344	0.396672	+	0.917960i	$0.370165\pi$
$500$	0	0
$501$	17.3717	0.776110
$502$	0	0
$503$	8.70054	0.387938	0.193969	−	0.981008i	$-0.437864\pi$
$503$	8.70054	0.387938	0.193969	+	0.981008i	$0.437864\pi$
$504$	0	0
$505$	10.3503	0.460581
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−33.3545	−1.47841	−0.739206	−	0.673480i	$-0.764799\pi$
$509$	−33.3545	−1.47841	−0.739206	+	0.673480i	$0.764799\pi$
$510$	0	0
$511$	−14.3074	−0.632923
$512$	0	0
$513$	−6.97858	−0.308112
$514$	0	0
$515$	18.7434	0.825932
$516$	0	0
$517$	1.22219	0.0537519
$518$	0	0
$519$	−7.95715	−0.349280
$520$	0	0
$521$	18.7005	0.819285	0.409643	−	0.912246i	$-0.365653\pi$
$521$	18.7005	0.819285	0.409643	+	0.912246i	$0.365653\pi$
$522$	0	0
$523$	4.00000	0.174908	0.0874539	−	0.996169i	$-0.472127\pi$
$523$	4.00000	0.174908	0.0874539	+	0.996169i	$0.472127\pi$
$524$	0	0
$525$	1.19656	0.0522221
$526$	0	0
$527$	18.3931	0.801217
$528$	0	0
$529$	−5.56825	−0.242098
$530$	0	0
$531$	−5.37169	−0.233112
$532$	0	0
$533$	−6.17513	−0.267475
$534$	0	0
$535$	−18.5682	−0.802775
$536$	0	0
$537$	15.5640	0.671638
$538$	0	0
$539$	−6.66273	−0.286984
$540$	0	0
$541$	−41.5296	−1.78550	−0.892749	−	0.450555i	$-0.851226\pi$
$541$	−41.5296	−1.78550	−0.892749	+	0.450555i	$0.851226\pi$
$542$	0	0
$543$	15.7820	0.677271
$544$	0	0
$545$	−8.39312	−0.359522
$546$	0	0
$547$	7.91431	0.338391	0.169196	−	0.985582i	$-0.445883\pi$
$547$	7.91431	0.338391	0.169196	+	0.985582i	$0.445883\pi$
$548$	0	0
$549$	12.5682	0.536400
$550$	0	0
$551$	−41.8715	−1.78378
$552$	0	0
$553$	2.13229	0.0906742
$554$	0	0
$555$	−7.78202	−0.330328
$556$	0	0
$557$	42.7005	1.80928	0.904640	−	0.426177i	$-0.140140\pi$
$557$	42.7005	1.80928	0.904640	+	0.426177i	$0.140140\pi$
$558$	0	0
$559$	9.95715	0.421143
$560$	0	0
$561$	7.38890	0.311960
$562$	0	0
$563$	−1.04706	−0.0441282	−0.0220641	−	0.999757i	$-0.507024\pi$
$563$	−1.04706	−0.0441282	−0.0220641	+	0.999757i	$0.507024\pi$
$564$	0	0
$565$	−7.95715	−0.334760
$566$	0	0
$567$	1.19656	0.0502507
$568$	0	0
$569$	16.7778	0.703362	0.351681	−	0.936120i	$-0.385610\pi$
$569$	16.7778	0.703362	0.351681	+	0.936120i	$0.385610\pi$
$570$	0	0
$571$	20.6111	0.862548	0.431274	−	0.902221i	$-0.358064\pi$
$571$	20.6111	0.862548	0.431274	+	0.902221i	$0.358064\pi$
$572$	0	0
$573$	−10.7434	−0.448811
$574$	0	0
$575$	−4.17513	−0.174115
$576$	0	0
$577$	1.38890	0.0578208	0.0289104	−	0.999582i	$-0.490796\pi$
$577$	1.38890	0.0578208	0.0289104	+	0.999582i	$0.490796\pi$
$578$	0	0
$579$	9.73917	0.404746
$580$	0	0
$581$	6.42754	0.266659
$582$	0	0
$583$	12.1751	0.504243
$584$	0	0
$585$	−1.00000	−0.0413449
$586$	0	0
$587$	0.935731	0.0386218	0.0193109	−	0.999814i	$-0.493853\pi$
$587$	0.935731	0.0386218	0.0193109	+	0.999814i	$0.493853\pi$
$588$	0	0
$589$	−20.7862	−0.856482
$590$	0	0
$591$	−9.56404	−0.393412
$592$	0	0
$593$	−0.478807	−0.0196622	−0.00983112	−	0.999952i	$-0.503129\pi$
$593$	−0.478807	−0.0196622	−0.00983112	+	0.999952i	$0.503129\pi$
$594$	0	0
$595$	−7.38890	−0.302916
$596$	0	0
$597$	−5.95715	−0.243810
$598$	0	0
$599$	−29.0852	−1.18839	−0.594195	−	0.804321i	$-0.702529\pi$
$599$	−29.0852	−1.18839	−0.594195	+	0.804321i	$0.702529\pi$
$600$	0	0
$601$	11.4318	0.466311	0.233155	−	0.972439i	$-0.425095\pi$
$601$	11.4318	0.466311	0.233155	+	0.972439i	$0.425095\pi$
$602$	0	0
$603$	−9.37169	−0.381645
$604$	0	0
$605$	9.56825	0.389005
$606$	0	0
$607$	−27.9143	−1.13301	−0.566503	−	0.824059i	$-0.691704\pi$
$607$	−27.9143	−1.13301	−0.566503	+	0.824059i	$0.691704\pi$
$608$	0	0
$609$	7.17935	0.290922
$610$	0	0
$611$	1.02142	0.0413223
$612$	0	0
$613$	−4.65394	−0.187971	−0.0939855	−	0.995574i	$-0.529961\pi$
$613$	−4.65394	−0.187971	−0.0939855	+	0.995574i	$0.529961\pi$
$614$	0	0
$615$	6.17513	0.249005
$616$	0	0
$617$	−15.9572	−0.642411	−0.321205	−	0.947010i	$-0.604088\pi$
$617$	−15.9572	−0.642411	−0.321205	+	0.947010i	$0.604088\pi$
$618$	0	0
$619$	−1.02142	−0.0410545	−0.0205272	−	0.999789i	$-0.506534\pi$
$619$	−1.02142	−0.0410545	−0.0205272	+	0.999789i	$0.506534\pi$
$620$	0	0
$621$	−4.17513	−0.167542
$622$	0	0
$623$	12.1751	0.487786
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	−8.35027	−0.333478
$628$	0	0
$629$	48.0550	1.91608
$630$	0	0
$631$	20.4998	0.816083	0.408041	−	0.912963i	$-0.366212\pi$
$631$	20.4998	0.816083	0.408041	+	0.912963i	$0.366212\pi$
$632$	0	0
$633$	−23.9143	−0.950508
$634$	0	0
$635$	−10.3931	−0.412438
$636$	0	0
$637$	−5.56825	−0.220622
$638$	0	0
$639$	5.19656	0.205573
$640$	0	0
$641$	38.2646	1.51136	0.755680	−	0.654941i	$-0.227307\pi$
$641$	38.2646	1.51136	0.755680	+	0.654941i	$0.227307\pi$
$642$	0	0
$643$	33.1109	1.30577	0.652883	−	0.757459i	$-0.273560\pi$
$643$	33.1109	1.30577	0.652883	+	0.757459i	$0.273560\pi$
$644$	0	0
$645$	−9.95715	−0.392063
$646$	0	0
$647$	16.9614	0.666820	0.333410	−	0.942782i	$-0.391801\pi$
$647$	16.9614	0.666820	0.333410	+	0.942782i	$0.391801\pi$
$648$	0	0
$649$	−6.42754	−0.252303
$650$	0	0
$651$	3.56404	0.139686
$652$	0	0
$653$	−19.1709	−0.750216	−0.375108	−	0.926981i	$-0.622394\pi$
$653$	−19.1709	−0.750216	−0.375108	+	0.926981i	$0.622394\pi$
$654$	0	0
$655$	−6.39312	−0.249800
$656$	0	0
$657$	−11.9572	−0.466493
$658$	0	0
$659$	−37.8715	−1.47526	−0.737631	−	0.675204i	$-0.764056\pi$
$659$	−37.8715	−1.47526	−0.737631	+	0.675204i	$0.764056\pi$
$660$	0	0
$661$	24.3931	0.948782	0.474391	−	0.880314i	$-0.342668\pi$
$661$	24.3931	0.948782	0.474391	+	0.880314i	$0.342668\pi$
$662$	0	0
$663$	6.17513	0.239822
$664$	0	0
$665$	8.35027	0.323810
$666$	0	0
$667$	−25.0508	−0.969971
$668$	0	0
$669$	−2.62831	−0.101616
$670$	0	0
$671$	15.0386	0.580560
$672$	0	0
$673$	−21.1281	−0.814428	−0.407214	−	0.913333i	$-0.633500\pi$
$673$	−21.1281	−0.814428	−0.407214	+	0.913333i	$0.633500\pi$
$674$	0	0
$675$	1.00000	0.0384900
$676$	0	0
$677$	−15.3973	−0.591767	−0.295884	−	0.955224i	$-0.595614\pi$
$677$	−15.3973	−0.591767	−0.295884	+	0.955224i	$0.595614\pi$
$678$	0	0
$679$	−2.18356	−0.0837972
$680$	0	0
$681$	−15.7648	−0.604109
$682$	0	0
$683$	−30.0722	−1.15068	−0.575341	−	0.817914i	$-0.695131\pi$
$683$	−30.0722	−1.15068	−0.575341	+	0.817914i	$0.695131\pi$
$684$	0	0
$685$	−16.7434	−0.639732
$686$	0	0
$687$	8.74338	0.333581
$688$	0	0
$689$	10.1751	0.387642
$690$	0	0
$691$	8.14950	0.310022	0.155011	−	0.987913i	$-0.450459\pi$
$691$	8.14950	0.310022	0.155011	+	0.987913i	$0.450459\pi$
$692$	0	0
$693$	1.43175	0.0543877
$694$	0	0
$695$	5.78202	0.219325
$696$	0	0
$697$	−38.1323	−1.44436
$698$	0	0
$699$	−2.17513	−0.0822712
$700$	0	0
$701$	−28.6921	−1.08369	−0.541843	−	0.840480i	$-0.682273\pi$
$701$	−28.6921	−1.08369	−0.541843	+	0.840480i	$0.682273\pi$
$702$	0	0
$703$	−54.3074	−2.04824
$704$	0	0
$705$	−1.02142	−0.0384690
$706$	0	0
$707$	−12.3847	−0.465774
$708$	0	0
$709$	12.3074	0.462215	0.231108	−	0.972928i	$-0.425765\pi$
$709$	12.3074	0.462215	0.231108	+	0.972928i	$0.425765\pi$
$710$	0	0
$711$	1.78202	0.0668310
$712$	0	0
$713$	−12.4360	−0.465730
$714$	0	0
$715$	−1.19656	−0.0447487
$716$	0	0
$717$	−2.80344	−0.104696
$718$	0	0
$719$	28.7862	1.07355	0.536773	−	0.843727i	$-0.319643\pi$
$719$	28.7862	1.07355	0.536773	+	0.843727i	$0.319643\pi$
$720$	0	0
$721$	−22.4275	−0.835245
$722$	0	0
$723$	−6.00000	−0.223142
$724$	0	0
$725$	6.00000	0.222834
$726$	0	0
$727$	−34.3931	−1.27557	−0.637785	−	0.770214i	$-0.720149\pi$
$727$	−34.3931	−1.27557	−0.637785	+	0.770214i	$0.720149\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	61.4868	2.27417
$732$	0	0
$733$	−29.0042	−1.07129	−0.535647	−	0.844442i	$-0.679932\pi$
$733$	−29.0042	−1.07129	−0.535647	+	0.844442i	$0.679932\pi$
$734$	0	0
$735$	5.56825	0.205388
$736$	0	0
$737$	−11.2138	−0.413065
$738$	0	0
$739$	6.27804	0.230941	0.115471	−	0.993311i	$-0.463162\pi$
$739$	6.27804	0.230941	0.115471	+	0.993311i	$0.463162\pi$
$740$	0	0
$741$	−6.97858	−0.256364
$742$	0	0
$743$	−12.2352	−0.448866	−0.224433	−	0.974490i	$-0.572053\pi$
$743$	−12.2352	−0.448866	−0.224433	+	0.974490i	$0.572053\pi$
$744$	0	0
$745$	−15.3461	−0.562236
$746$	0	0
$747$	5.37169	0.196540
$748$	0	0
$749$	22.2180	0.811827
$750$	0	0
$751$	28.8757	1.05369	0.526844	−	0.849962i	$-0.323375\pi$
$751$	28.8757	1.05369	0.526844	+	0.849962i	$0.323375\pi$
$752$	0	0
$753$	23.9143	0.871486
$754$	0	0
$755$	−8.58546	−0.312457
$756$	0	0
$757$	30.3503	1.10310	0.551550	−	0.834142i	$-0.314037\pi$
$757$	30.3503	1.10310	0.551550	+	0.834142i	$0.314037\pi$
$758$	0	0
$759$	−4.99579	−0.181336
$760$	0	0
$761$	−27.1709	−0.984945	−0.492473	−	0.870328i	$-0.663907\pi$
$761$	−27.1709	−0.984945	−0.492473	+	0.870328i	$0.663907\pi$
$762$	0	0
$763$	10.0428	0.363575
$764$	0	0
$765$	−6.17513	−0.223262
$766$	0	0
$767$	−5.37169	−0.193961
$768$	0	0
$769$	−38.3503	−1.38295	−0.691473	−	0.722402i	$-0.743038\pi$
$769$	−38.3503	−1.38295	−0.691473	+	0.722402i	$0.743038\pi$
$770$	0	0
$771$	−19.9572	−0.718739
$772$	0	0
$773$	−50.2646	−1.80789	−0.903946	−	0.427647i	$-0.859342\pi$
$773$	−50.2646	−1.80789	−0.903946	+	0.427647i	$0.859342\pi$
$774$	0	0
$775$	2.97858	0.106994
$776$	0	0
$777$	9.31163	0.334053
$778$	0	0
$779$	43.0937	1.54399
$780$	0	0
$781$	6.21798	0.222497
$782$	0	0
$783$	6.00000	0.214423
$784$	0	0
$785$	−2.78623	−0.0994448
$786$	0	0
$787$	13.2860	0.473595	0.236797	−	0.971559i	$-0.423902\pi$
$787$	13.2860	0.473595	0.236797	+	0.971559i	$0.423902\pi$
$788$	0	0
$789$	−8.00000	−0.284808
$790$	0	0
$791$	9.52119	0.338535
$792$	0	0
$793$	12.5682	0.446312
$794$	0	0
$795$	−10.1751	−0.360875
$796$	0	0
$797$	9.82487	0.348015	0.174007	−	0.984744i	$-0.444328\pi$
$797$	9.82487	0.348015	0.174007	+	0.984744i	$0.444328\pi$
$798$	0	0
$799$	6.30742	0.223141
$800$	0	0
$801$	10.1751	0.359521
$802$	0	0
$803$	−14.3074	−0.504898
$804$	0	0
$805$	4.99579	0.176078
$806$	0	0
$807$	−2.35027	−0.0827334
$808$	0	0
$809$	−9.91431	−0.348569	−0.174284	−	0.984695i	$-0.555761\pi$
$809$	−9.91431	−0.348569	−0.174284	+	0.984695i	$0.555761\pi$
$810$	0	0
$811$	36.5855	1.28469	0.642345	−	0.766416i	$-0.277962\pi$
$811$	36.5855	1.28469	0.642345	+	0.766416i	$0.277962\pi$
$812$	0	0
$813$	−10.9786	−0.385036
$814$	0	0
$815$	8.76060	0.306870
$816$	0	0
$817$	−69.4868	−2.43103
$818$	0	0
$819$	1.19656	0.0418111
$820$	0	0
$821$	−34.4741	−1.20316	−0.601578	−	0.798814i	$-0.705461\pi$
$821$	−34.4741	−1.20316	−0.601578	+	0.798814i	$0.705461\pi$
$822$	0	0
$823$	−13.2566	−0.462097	−0.231048	−	0.972942i	$-0.574216\pi$
$823$	−13.2566	−0.462097	−0.231048	+	0.972942i	$0.574216\pi$
$824$	0	0
$825$	1.19656	0.0416588
$826$	0	0
$827$	−28.1495	−0.978854	−0.489427	−	0.872044i	$-0.662794\pi$
$827$	−28.1495	−0.978854	−0.489427	+	0.872044i	$0.662794\pi$
$828$	0	0
$829$	16.3418	0.567576	0.283788	−	0.958887i	$-0.408409\pi$
$829$	16.3418	0.567576	0.283788	+	0.958887i	$0.408409\pi$
$830$	0	0
$831$	1.21377	0.0421052
$832$	0	0
$833$	−34.3847	−1.19136
$834$	0	0
$835$	−17.3717	−0.601172
$836$	0	0
$837$	2.97858	0.102955
$838$	0	0
$839$	30.3675	1.04840	0.524201	−	0.851595i	$-0.324364\pi$
$839$	30.3675	1.04840	0.524201	+	0.851595i	$0.324364\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	0	0
$843$	11.9572	0.411826
$844$	0	0
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−11.4490	−0.393391
$848$	0	0
$849$	29.8715	1.02519
$850$	0	0
$851$	−32.4910	−1.11378
$852$	0	0
$853$	−42.1407	−1.44287	−0.721435	−	0.692482i	$-0.756517\pi$
$853$	−42.1407	−1.44287	−0.721435	+	0.692482i	$0.756517\pi$
$854$	0	0
$855$	6.97858	0.238662
$856$	0	0
$857$	−2.17513	−0.0743012	−0.0371506	−	0.999310i	$-0.511828\pi$
$857$	−2.17513	−0.0743012	−0.0371506	+	0.999310i	$0.511828\pi$
$858$	0	0
$859$	−18.5682	−0.633541	−0.316770	−	0.948502i	$-0.602598\pi$
$859$	−18.5682	−0.633541	−0.316770	+	0.948502i	$0.602598\pi$
$860$	0	0
$861$	−7.38890	−0.251813
$862$	0	0
$863$	−33.7220	−1.14791	−0.573954	−	0.818887i	$-0.694591\pi$
$863$	−33.7220	−1.14791	−0.573954	+	0.818887i	$0.694591\pi$
$864$	0	0
$865$	7.95715	0.270551
$866$	0	0
$867$	21.1323	0.717690
$868$	0	0
$869$	2.13229	0.0723330
$870$	0	0
$871$	−9.37169	−0.317548
$872$	0	0
$873$	−1.82487	−0.0617623
$874$	0	0
$875$	−1.19656	−0.0404510
$876$	0	0
$877$	43.4868	1.46844	0.734222	−	0.678910i	$-0.237547\pi$
$877$	43.4868	1.46844	0.734222	+	0.678910i	$0.237547\pi$
$878$	0	0
$879$	0.777809	0.0262348
$880$	0	0
$881$	26.7005	0.899564	0.449782	−	0.893138i	$-0.351502\pi$
$881$	26.7005	0.899564	0.449782	+	0.893138i	$0.351502\pi$
$882$	0	0
$883$	−20.2990	−0.683116	−0.341558	−	0.939861i	$-0.610955\pi$
$883$	−20.2990	−0.683116	−0.341558	+	0.939861i	$0.610955\pi$
$884$	0	0
$885$	5.37169	0.180567
$886$	0	0
$887$	−36.0550	−1.21061	−0.605305	−	0.795994i	$-0.706949\pi$
$887$	−36.0550	−1.21061	−0.605305	+	0.795994i	$0.706949\pi$
$888$	0	0
$889$	12.4360	0.417089
$890$	0	0
$891$	1.19656	0.0400862
$892$	0	0
$893$	−7.12808	−0.238532
$894$	0	0
$895$	−15.5640	−0.520248
$896$	0	0
$897$	−4.17513	−0.139404
$898$	0	0
$899$	17.8715	0.596047
$900$	0	0
$901$	62.8328	2.09326
$902$	0	0
$903$	11.9143	0.396483
$904$	0	0
$905$	−15.7820	−0.524612
$906$	0	0
$907$	7.26504	0.241232	0.120616	−	0.992699i	$-0.461513\pi$
$907$	7.26504	0.241232	0.120616	+	0.992699i	$0.461513\pi$
$908$	0	0
$909$	−10.3503	−0.343297
$910$	0	0
$911$	6.65769	0.220579	0.110290	−	0.993899i	$-0.464822\pi$
$911$	6.65769	0.220579	0.110290	+	0.993899i	$0.464822\pi$
$912$	0	0
$913$	6.42754	0.212721
$914$	0	0
$915$	−12.5682	−0.415494
$916$	0	0
$917$	7.64973	0.252616
$918$	0	0
$919$	27.1831	0.896688	0.448344	−	0.893861i	$-0.352014\pi$
$919$	27.1831	0.896688	0.448344	+	0.893861i	$0.352014\pi$
$920$	0	0
$921$	−0.760597	−0.0250625
$922$	0	0
$923$	5.19656	0.171047
$924$	0	0
$925$	7.78202	0.255871
$926$	0	0
$927$	−18.7434	−0.615614
$928$	0	0
$929$	−15.3973	−0.505170	−0.252585	−	0.967575i	$-0.581281\pi$
$929$	−15.3973	−0.505170	−0.252585	+	0.967575i	$0.581281\pi$
$930$	0	0
$931$	38.8585	1.27353
$932$	0	0
$933$	23.1281	0.757179
$934$	0	0
$935$	−7.38890	−0.241643
$936$	0	0
$937$	1.12808	0.0368527	0.0184264	−	0.999830i	$-0.494134\pi$
$937$	1.12808	0.0368527	0.0184264	+	0.999830i	$0.494134\pi$
$938$	0	0
$939$	−33.9143	−1.10675
$940$	0	0
$941$	−30.1407	−0.982559	−0.491280	−	0.871002i	$-0.663471\pi$
$941$	−30.1407	−0.982559	−0.491280	+	0.871002i	$0.663471\pi$
$942$	0	0
$943$	25.7820	0.839578
$944$	0	0
$945$	−1.19656	−0.0389240
$946$	0	0
$947$	20.0294	0.650868	0.325434	−	0.945565i	$-0.394490\pi$
$947$	20.0294	0.650868	0.325434	+	0.945565i	$0.394490\pi$
$948$	0	0
$949$	−11.9572	−0.388146
$950$	0	0
$951$	9.64973	0.312914
$952$	0	0
$953$	−43.2259	−1.40023	−0.700113	−	0.714032i	$-0.746867\pi$
$953$	−43.2259	−1.40023	−0.700113	+	0.714032i	$0.746867\pi$
$954$	0	0
$955$	10.7434	0.347648
$956$	0	0
$957$	7.17935	0.232075
$958$	0	0
$959$	20.0344	0.646945
$960$	0	0
$961$	−22.1281	−0.713809
$962$	0	0
$963$	18.5682	0.598353
$964$	0	0
$965$	−9.73917	−0.313515
$966$	0	0
$967$	−57.6875	−1.85511	−0.927553	−	0.373691i	$-0.878092\pi$
$967$	−57.6875	−1.85511	−0.927553	+	0.373691i	$0.878092\pi$
$968$	0	0
$969$	−43.0937	−1.38437
$970$	0	0
$971$	19.5296	0.626735	0.313368	−	0.949632i	$-0.398543\pi$
$971$	19.5296	0.626735	0.313368	+	0.949632i	$0.398543\pi$
$972$	0	0
$973$	−6.91852	−0.221798
$974$	0	0
$975$	1.00000	0.0320256
$976$	0	0
$977$	−40.3074	−1.28955	−0.644774	−	0.764373i	$-0.723049\pi$
$977$	−40.3074	−1.28955	−0.644774	+	0.764373i	$0.723049\pi$
$978$	0	0
$979$	12.1751	0.389119
$980$	0	0
$981$	8.39312	0.267972
$982$	0	0
$983$	−32.2008	−1.02705	−0.513523	−	0.858076i	$-0.671660\pi$
$983$	−32.2008	−1.02705	−0.513523	+	0.858076i	$0.671660\pi$
$984$	0	0
$985$	9.56404	0.304736
$986$	0	0
$987$	1.22219	0.0389028
$988$	0	0
$989$	−41.5725	−1.32193
$990$	0	0
$991$	−26.4826	−0.841246	−0.420623	−	0.907235i	$-0.638189\pi$
$991$	−26.4826	−0.841246	−0.420623	+	0.907235i	$0.638189\pi$
$992$	0	0
$993$	−15.3288	−0.486446
$994$	0	0
$995$	5.95715	0.188854
$996$	0	0
$997$	35.1365	1.11278	0.556392	−	0.830920i	$-0.312185\pi$
$997$	35.1365	1.11278	0.556392	+	0.830920i	$0.312185\pi$
$998$	0	0
$999$	7.78202	0.246212

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3120.2.a.bj.1.2		3
3.2	odd	2		9360.2.a.dd.1.2		3
4.3	odd	2		195.2.a.e.1.1	✓	3
12.11	even	2		585.2.a.n.1.3		3
20.3	even	4		975.2.c.i.274.5		6
20.7	even	4		975.2.c.i.274.2		6
20.19	odd	2		975.2.a.o.1.3		3
28.27	even	2		9555.2.a.bq.1.1		3
52.51	odd	2		2535.2.a.bc.1.3		3
60.23	odd	4		2925.2.c.w.2224.2		6
60.47	odd	4		2925.2.c.w.2224.5		6
60.59	even	2		2925.2.a.bh.1.1		3
156.155	even	2		7605.2.a.bx.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.a.e.1.1	✓	3	4.3	odd	2
585.2.a.n.1.3		3	12.11	even	2
975.2.a.o.1.3		3	20.19	odd	2
975.2.c.i.274.2		6	20.7	even	4
975.2.c.i.274.5		6	20.3	even	4
2535.2.a.bc.1.3		3	52.51	odd	2
2925.2.a.bh.1.1		3	60.59	even	2
2925.2.c.w.2224.2		6	60.23	odd	4
2925.2.c.w.2224.5		6	60.47	odd	4
3120.2.a.bj.1.2		3	1.1	even	1	trivial
7605.2.a.bx.1.1		3	156.155	even	2
9360.2.a.dd.1.2		3	3.2	odd	2
9555.2.a.bq.1.1		3	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 3120 = 2^{4} \cdot 3 \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3120.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.19656 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.00000 q^{5} +1.19656 q^{7} +1.00000 q^{9} +1.19656 q^{11} +1.00000 q^{13} -1.00000 q^{15} +6.17513 q^{17} -6.97858 q^{19} +1.19656 q^{21} -4.17513 q^{23} +1.00000 q^{25} +1.00000 q^{27} +6.00000 q^{29} +2.97858 q^{31} +1.19656 q^{33} -1.19656 q^{35} +7.78202 q^{37} +1.00000 q^{39} -6.17513 q^{41} +9.95715 q^{43} -1.00000 q^{45} +1.02142 q^{47} -5.56825 q^{49} +6.17513 q^{51} +10.1751 q^{53} -1.19656 q^{55} -6.97858 q^{57} -5.37169 q^{59} +12.5682 q^{61} +1.19656 q^{63} -1.00000 q^{65} -9.37169 q^{67} -4.17513 q^{69} +5.19656 q^{71} -11.9572 q^{73} +1.00000 q^{75} +1.43175 q^{77} +1.78202 q^{79} +1.00000 q^{81} +5.37169 q^{83} -6.17513 q^{85} +6.00000 q^{87} +10.1751 q^{89} +1.19656 q^{91} +2.97858 q^{93} +6.97858 q^{95} -1.82487 q^{97} +1.19656 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^5 - q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{5} - q^{7} + 3 q^{9} - q^{11} + 3 q^{13} - 3 q^{15} - q^{17} - 6 q^{19} - q^{21} + 7 q^{23} + 3 q^{25} + 3 q^{27} + 18 q^{29} - 6 q^{31} - q^{33} + q^{35} + 13 q^{37} + 3 q^{39} + q^{41} - 3 q^{45} + 18 q^{47} + 12 q^{49} - q^{51} + 11 q^{53} + q^{55} - 6 q^{57} + 8 q^{59} + 9 q^{61} - q^{63} - 3 q^{65} - 4 q^{67} + 7 q^{69} + 11 q^{71} - 6 q^{73} + 3 q^{75} + 33 q^{77} - 5 q^{79} + 3 q^{81} - 8 q^{83} + q^{85} + 18 q^{87} + 11 q^{89} - q^{91} - 6 q^{93} + 6 q^{95} - 25 q^{97} - q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^5 - q^7 + 3 * q^9 - q^11 + 3 * q^13 - 3 * q^15 - q^17 - 6 * q^19 - q^21 + 7 * q^23 + 3 * q^25 + 3 * q^27 + 18 * q^29 - 6 * q^31 - q^33 + q^35 + 13 * q^37 + 3 * q^39 + q^41 - 3 * q^45 + 18 * q^47 + 12 * q^49 - q^51 + 11 * q^53 + q^55 - 6 * q^57 + 8 * q^59 + 9 * q^61 - q^63 - 3 * q^65 - 4 * q^67 + 7 * q^69 + 11 * q^71 - 6 * q^73 + 3 * q^75 + 33 * q^77 - 5 * q^79 + 3 * q^81 - 8 * q^83 + q^85 + 18 * q^87 + 11 * q^89 - q^91 - 6 * q^93 + 6 * q^95 - 25 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more